Capacitors and Capacitance

CAPACITOR

A capacitor is a two-terminal electrical device that can store energy in the form of an electric charge. It consists of two electrical conductors that are separated by a distance.  The space between the conductors may be filled by vacuum or with an insulating material known as a dielectric. The ability of the capacitor to store charges is known as capacitance.

Capacitors store energy by holding apart pairs of opposite charges. The simplest design for a capacitor is a parallel plate, which consists of two metal plates with a gap between them. But, different types of capacitors are manufactured in many forms, styles, lengths, girths, and materials.

Capacitor Working

For proof, let's look at the most basic structure of a capacitor – the parallel plate capacitor. It consists of two parallel plates separated by a dielectric. When we connect a DC voltage source across the capacitor, one plate is connected to the positive end (plate I) and the other to the negative end (plate II). When battery potential is applied across the capacitor, plate I become positive with respect to plate II. The current tries to flow through the capacitor at the steady-state condition from its positive plate to the negative plate. But it cannot flow due to the separation of the plates by insulating material.

An electric field appears across the capacitor. The positive plate (plate I) collects positive charges from the battery and the negative plate (plate II) accumulates negative charges from the battery.  After a point, the capacitor holds the maximum amount of is proportional to its voltage times its capacitance. This period is called the capacitor charging time.

When the battery is removed from the capacitor, the two plates hold a negative and positive charge for a certain time. Thus, the capacitor acts as a source of electrical energy.

When those plates are connected to a load,  current flows into the load from plate I to plate II until all the charges are dissipated from both plates. This period is called the time of discharge of the capacitor.

Determine the Value of Capacitance

The conducting plates have some charges Q1 and Q2 (Usually, if one plate has +q, the other has –q charge). The electric field in the region between the plates depends on the charge given to the conducting plates. We also know that potential difference (V) is directly proportional to the electric field hence we can say,


This constant of proportionality is known as the capacitance of the capacitor.

Capacitance is the ratio of the change in the electric charge of a system to the corresponding change in its electric potential.

Energy Stored in a Capacitor

 

The energy stored in a capacitor is nothing but the electric potential energy and is related to the voltage and charge on the capacitor. If the capacitance of a conductor is C, then it is initially uncharged and it acquires a potential difference V when connected to a battery. If q is the charge on the plate at that time, then

Applications of Capacitor Energy

Following are a few applications of capacitor energy:

  • A defibrillator that is used to correct abnormal heart rhythm delivers a large charge in a short burst to a person’s heart. Applying large shocks of electric current can stop the arrhythmia and allow the body’s natural pacemaker to resume its normal rhythm. A defibrillator uses the energy stored in the capacitor.
  • The audio equipment, uninterruptible power supplies, camera flashes, pulsed loads such as magnetic coils and lasers use the energy stored in the capacitors.
  • Super capacitors are capable of storing a large amount of energy and can offer new technological possibilities.

Standard Units of Capacitance

The basic unit of capacitance is Farad. But, Farad is a large unit for practical tasks. Hence, capacitance is usually measured in the sub-units of Farads, such as micro-farads (µF) or pico-farads (pF).

Most of the electrical and electronic applications are covered by the following standard unit (SI) prefixes for easy calculations:

·        1 mF (millifarad) = 10−3 F

·        1 μF (microfarad) =10−6 F

·        1 nF (nanofarad) = 10−9 F

·        1 pF (picofarad) = 10−12 F

Parallel Plate Capacitor

Parallel Plate Capacitors are formed by an arrangement of electrodes and insulating material or dielectric. A parallel plate capacitor can only store a finite amount of energy before dielectric breakdown occurs. It can be defined as:

When two parallel plates are connected across a battery, the plates are charged and an electric field is established between them, and this setup is known as the parallel plate capacitor.

Parallel Plate Capacitor Formula

The direction of the electric field is defined as the direction in which the positive test charge would flow. Capacitance is the limitation of the body to store the electric charge. Every capacitor has its capacitance. The typical parallel-plate capacitor consists of two metallic plates of area A, separated by the distance d.

The parallel plate capacitor formula is given by:

Where,

  • ϵo is the permittivity of space (8.854 × 10−12 F/m)
  • k is the relative permittivity of dielectric material
  • d is the separation between the plates
  • A is the area of plates

Parallel Plate Capacitor Derivation

The figure below depicts a parallel plate capacitor. We can see two large plates placed parallel to each other at a small distance d. The distance between the plates is filled with a dielectric medium as shown by the dotted array. The two plates carry an equal and opposite charge.

Here, we see that the first plate carries a charge +Q and the second carries a charge –Q. The area of each of the plates is A and the distance between these two plates is d. The distance d is much smaller than the area of the plates and we can write d<<A, thus the effect of the plates are considered as infinite plane sheets and the electric field generated by them is treated as that equal to the electric field generated by an infinite plane sheet of uniform surface charge density. As the total charge on plate 1 is Q and the area of the plate is A, the surface charge density can be given as

Similarly, for plate 2 with a total charge equal to –Q and area A, the surface charge density can be given as,

 

We divide the regions around the parallel plate capacitor into three parts, with region 1 being the area left to the first plate, region 2 being the area between the two plates and region 3 being the area to the right of plate 2.

Let us calculate the electric field in the region around a parallel plate capacitor.

Region I: The magnitude of the electric field due to both the infinite plane sheets I and II is the same at any point in this region, but the direction is opposite to each other, the two forces cancel each other and the overall electric field can be given as,

Capacitance of a Spherical Capacitor

Spherical capacitors consist of two concentric conducting spherical shells of radii  R1 and R2. The shells are given equal and opposite charges +Q and –Q respectively. The electric field between shells is directed radially outward. The magnitude of the field can be obtained by applying Gauss law over a spherical Gaussian surface of radius r concentric with the shells.

Factors Affecting Capacitance

Dielectric

The effect of dielectric on capacitance is that the greater the permittivity of the dielectric, the greater the capacitance, likewise lesser the permittivity of the dielectric the lesser is the capacitance. Some materials offer less opposition to the field flux for a given amount of field force. Materials with greater permittivity allow more field flux. Hence greater charge is collected.

Plate Spacing

The effect of spacing on the capacitance is that it is inversely proportional to the distance between the plates. Mathematically it is given as:

C α 1d

Area of the Plates

The effect of the area of the plate is that the capacitance is directly proportional to the area. The larger the plate area, the more the capacitance value. Mathematically it is given as:

αA

Effect of Dielectric on Capacitance

To know the effect of dielectric on capacitance let us consider a simple capacitor with parallel plates of area A, separated by a distance d, we can see that the charge on each plate is +Q and –Q for a capacitor with charge Q. As the area of the plate is A, the corresponding charge density can be given as ±σ. Where,

Let us consider another capacitor with the exact specifications as taken before. Let us insert a dielectric between the plates such that it fully occupies the space between the plates. As the dielectric enters the field between the plates, it gets polarized by the field, and the charges get arranged such that they act as two charged sheets with a surface charge density of σp and – σp, as shown in the figure below.

The net surface charge density then becomes equivalent to ±(σ – σp).

The potential difference across the capacitor can thus be given as,

 Applications of Capacitors

Capacitors for Energy Storage

Since the late 18th century, capacitors have been used to store electrical energy. Individual capacitors do not hold much energy, providing only enough power for electronic devices during temporary power outages or when they need additional power. Many applications use capacitors as energy sources, and a few of them are as follows:

  • Audio equipment
  • Camera Flashes
  • Power supplies
  • Magnetic coils
  • Lasers

Super capacitors are capacitors that have high capacitances up to 2 kF. These capacitors store large amounts of energy and offer new technological possibilities in areas such as electric cars, regenerative braking in the automotive industry and industrial electrical motors, computer memory backup during power loss, and many others.

Capacitors for Power Conditioning

One of the important applications of capacitors is the conditioning of power supplies. Capacitors allow only AC signals to pass when they are charged, blocking DC signals. This capacitor effect is used in separating or decoupling different parts of electrical circuits to reduce noise as a result of improving efficiency. Capacitors are also used in utility substations to counteract inductive loading introduced by transmission lines.

Capacitors as Sensors

Capacitors are used as sensors to measure a variety of things including humidity, mechanical strain, and fuel levels. Two aspects of capacitor construction are used in the sensing application – the distance between the parallel plates and the material between them. The former detects mechanical changes such as acceleration and pressure, and the latter is used in sensing air humidity.

Capacitors for Signal Processing

There are advanced applications****** of capacitors in information technology. Capacitors are used by Dynamic Random Access Memory (DRAM) devices to represent binary information as bits. Capacitors are also used in conjunction with inductors to tune circuits to particular frequencies, an effect exploited by radio receivers, speakers, and analog equalizers.